# Subgroup and quotient

**Algebra**Level 5

Let \(G\) be a group and \(N\) a normal subgroup. Which of the following statements is/are always true?

**I.** If \(N\) is finite and \(G/N\) is finite, then \(G\) is finite.

**II.** If \(N\) is finite and cyclic and \(G/N\) is finite and cyclic, then \(G\) is finite and cyclic.

**III.** If \(N\) is abelian and \(G/N\) is abelian, then \(G\) is abelian.

**Notation:**

A finite cyclic group is a group that is isomorphic to \( {\mathbb Z}_n,\) the integers mod \(n,\) for some \(n.\)

An abelian group is a group whose operation is commutative: \(x * y = y * x\) for all \(x,y \in G.\)